Algorithmic View of Online Prize-collecting Optimization Problems
Christine Markarian
1
and Abdul Nasser El-Kassar
2
1
Department of Engineering and Information Technology, University of Dubai, U.A.E.
2
Department of Information Technology and Operations Management, Lebanese American University, Lebanon
Keywords:
Prize-collecting, Facility Location Planning, Decision Making, Online Algorithms, Competitive Analysis.
Abstract:
Online algorithms have been a cornerstone of research in network design problems. Unlike in classical offline
algorithms, the input to an online algorithm is revealed in portions over time and the online algorithm reacts
to each portion while targeting a given optimization goal. Online algorithms are deployed in real-world opti-
mization problems in which provably good decisions are expected in the present without knowing the future.
In this paper, we consider a well-established branch of online optimization problems, known as online prize-
collecting problems, in which the online algorithm may reject some input portions at the cost of paying an
associated penalty. These appear in business applications in which a company decides to lose some customers
by paying an associated penalty. Particularly, we study the online prize-collecting variants of three well-known
optimization problems: Connected Dominating Set, Vertex Cover, and Non-metric Facility Location, namely,
Online Prize-collecting Connected Dominating Set (OPC-CDS), Online Prize-collecting Vertex Cover (OPC-
VC), and Online Prize-collecting Non-metric Facility Location (OPC-NFL), respectively. We propose the first
online algorithms for these variants and evaluate them using competitive analysis, the standard framework to
measure online algorithms, in which the online algorithm is measured against the optimal offline algorithm
that knows the entire input sequence in advance and is optimal.
1 INTRODUCTION
Many real-world optimization problems are online
in nature, requiring smart decisions to be made im-
mediately, even when future information is not fully
known. The challenge is then to make decisions that
will cause as few regrets as possible in the future.
Classically, most optimization problems have been
modeled in an offline setting, in which the entire in-
put sequence is assumed to be known to the decision
maker in advance. In many real-world scenarios, hav-
ing access to future input sequences is most of the
time hard or even impossible. Over the past decades,
online algorithms have gained a lot of popularity in
such scenarios, since they can provide performance
guarantee to the decision maker. That is, decisions
made by online algorithms are measured against op-
timal decisions that are made by the assumption of
knowing the entire input sequence in advance. Hence,
a decision maker would know how good or bad the
decision is at the time it is made. Moreover, it would
allow decision makers to know and try to achieve the
best provably possible decision achievable by any de-
cision maker. Unlike in classical offline models, the
input to an online algorithm is revealed in portions
over time and the online algorithm reacts to each por-
tion, as it targets a given optimization goal against the
entire input sequence.
We adopt the competitive analysis framework to
evaluate the online algorithms (Sleator and Tarjan,
1985). In this framework, the performance of the
online algorithm is measured against the optimal of-
fline algorithm, that knows the entire input sequence
in advance and is optimal, in the worst case. Given
an input sequence σ, let C
A
(σ) and C
OPT
(σ) de-
note the cost incurred by an online algorithm A, pos-
sibly randomized, and an optimal offline algorithm
OPT , respectively. A has competitive ratio c or is
c-competitive if there exists a constant α such that
C
A
(σ) c · C
OPT
(σ) + α for all input sequences σ.
We assume the oblivious adversarial model, in which
the adversary specifies all of the input at the begin-
ning and does not know the random outcomes of the
algorithm.
In this paper, we consider a well-established
branch of online optimization problems known as on-
line prize-collecting problems, in which the online al-
gorithm may reject some input portions at the cost of
paying an associated penalty. These appear in net-
work planning applications for service providers, in
744
Markarian, C. and El-Kassar, A.
Algorithmic View of Online Prize-collecting Optimization Problems.
DOI: 10.5220/0010471507440751
In Proceedings of the 23rd International Conference on Enterprise Information Systems (ICEIS 2021) - Volume 1, pages 744-751
ISBN: 978-989-758-509-8
Copyright
c
2021 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
which a company may decide to lose some customers
by paying a corresponding penalty in the revenue.
Since (Qian and Williamson, 2011) introduced the
online prize collecting framework, many graph opti-
mization problems, such as variants of Steiner prob-
lems and metric Facility Location, were defined in
this model (see (Felice et al., 2015; Hajiaghayi et al.,
2013; Markarian, 2018)). It is worth noting that the
online prize-collecting model is a generalization of
the online model in which all penalties are set to in-
finity.
2 OUR CONTRIBUTION
In this paper, we study the online prize-collecting
variants of three classical optimization problems:
Connected Dominating Set, Vertex Cover, and Non-
metric Facility Location.
2.1 Online Prize-collecting Connected
Dominating Set
Consider an advertising company trying to reach po-
tential customers in a social network for the purpose
of advertising a certain product or service. The com-
pany assumes a customer is reached either if he has
direct access to the ad or one of his friends has. Ev-
ery now and then, a number of potential customers are
announced. These are people who would likely be in-
terested in the ad. The point is to send the ad to as few
people as possible so as to reduce costs, while reach-
ing as many people as possible. To achieve this, the
company wishes to find a group of connected people
who can spread the message to each other and to the
rest of the potential customers through the word-of-
mouth effect. The company may ignore some of the
potential customers, by paying an associated penalty.
The goal is to minimize the total costs of sending ads
and penalties.
From an algorithmic perspective, this scenario can
be formalized as the Online Prize-collecting Con-
nected Dominating Set problem (OPC-CDS). OPC-
CDS is the online prize-collecting variant of Con-
nected Dominating Set. It generalizes the Online
Set Cover problem (OSC) introduced by (Alon et al.,
2003), defined as follows.
Definition 1. (OSC) Given a universe U of elements
and a collection S of subsets of U, each associated
with a cost. A subset D U of elements arrives over
time and OSC asks to find a minimum cost of subsets
C S that cover all elements in D.
(Korman, 2005) gave a lower bound of (log mlogn)
on the competitive ratio of any online polynomial-
time randomized algorithm for OSC, under the as-
sumption that NP 6⊆ BPP, where m is the number of
subsets and n is the number of elements. This implies
a lower bound of (log
2
n) on the competitive ratio of
any randomized polynomial-time algorithm for OPC-
CDS, where n is the number of nodes. OPC-CDS is
defined as follows.
Definition 2. (OPC-CDS) Given an undirected con-
nected graph G = (V, E) with |V | = n, node-weight
function w : V R
+
, and penalty-cost function p :
V R
+
. A sequence of disjoint subsets of V arrives
over time. A subset S V serves as a connected dom-
inating set of a given subset D V if every node in
D is either in S or has an adjacent node in S, and the
subgraph induced by S is connected in G. In each step
t, a subset D
t
V arrives; for each u D
t
, OPC-CDS
asks to either pay the penalty p
u
of u or add u to a sub-
set D
0
t
D
t
that is served by a connected dominating
set, at time t. The goal is to minimize the total weight
of the connected dominating set constructed and the
total penalties paid.
To the best of our knowledge, no online algo-
rithm with non-trivial competitive ratio exists for
OPC-CDS. A special case of OPC-CDS, in the online
model, namely, the Online Connected Dominating Set
problem (OCDS), was introduced by (Hamann et al.,
2018), in the context of modern robotic warehouses.
Unlike in this paper, (Hamann et al., 2018) studied a
special case in which all node-weights are uniform.
Hence, their approach cannot be applied to our prob-
lem.
In this paper, we propose the first online algorithm
for OPC-CDS, with O(
w
max
w
min
log
2
n)-competitive ratio,
where n is the number of nodes, w
max
is the maximum
node weight, and w
min
is the minimum node weight.
Our algorithm is randomized and makes use of the
deterministic algorithm of (Alon et al., 2003) for the
Online Set Cover problem (OSC), defined earlier, and
the randomized algorithm of (Hajiaghayi et al., 2014)
for the Online Node-weighted Steiner Tree problem
(OPC-NWST).
2.2 Online Prize-collecting Vertex
Cover
Consider the advertising company described earlier.
For some ads, the company wishes to assure that the
ad is reached to the customers without relying on the
word-of-mouth effect. This means, it assumes a cus-
tomer is influenced only through close friends. Every
now and then, a number of friendships are announced.
Algorithmic View of Online Prize-collecting Optimization Problems
745
These are pairs of people who would likely influence
each other. If one of them receives the ad, the other
is assumed to have been reached. Moreover, the com-
pany may wish, as before, to ignore some friendships,
at the cost of paying a penalty. The goal is to mini-
mize the total costs of sending ads and penalties.
This scenario can be modeled as the Online Prize-
collecting Vertex Cover problem (OPC-VC). OPC-
VC is the online prize-collecting variant of Vertex
Cover and is defined as follows.
Definition 3. (OPC-VC) Given an undirected graph
G = (V, E) with |V | = n and node-weight function w :
V R
+
. A sequence of edges, each associated with
a penalty, arrives over time. In each step, an edge
arrives; OPC-VC asks to output a set S of nodes, such
that each edge has at least one of its endpoints in S or
its penalty is paid, at the current step. The goal is to
minimize the total weight of S and the total penalties
paid.
To the best of our knowledge, no online algorithm
with non-trivial competitive ratio exists for OPC-VC.
A special case of OPC-VC is the Online Vertex Cover
problem (OVC). For the unweighted variant of OVC,
in which all node weights are uniform, there is a sim-
ple greedy algorithm with 2-competitive ratio. As
soon as an edge arrives, if it is not covered, this al-
gorithm adds both of its endpoints to the solution.
In this paper, we propose the first online algorithm
for OPC-VC, with 3-competitive ratio. Our algorithm
is deterministic and is based on a simple classical
primal-dual approach.
2.3 Online Prize-collecting Non-metric
Facility Location
Assuming the word-of-mouth effect, the advertising
company now wishes to send the ad to very few peo-
ple so as every potential customer announced is not
very far from one of these people. As before, some
potential customers may be ignored at the cost of
paying a penalty. The goal is to minimize the total
costs of sending ads, penalties, and the total distances
from the potential customers to the nearest people
who have received the ad.
This scenario can be formulated as the Online
Prize-collecting Non-metric Facility Location prob-
lem (OPC-NFL). OPC-NFL is the online prize-
collecting variant of Non-metric Facility Location and
is defined as follows.
Definition 4. (OPC-NFL) Given a complete bipartite
graph G = ((F D), E), where F is the set of facilities
that may be opened and D is the set of clients arriv-
ing over time, an edge-weight function w : E R
+
,
a facility-opening-cost function f : F R
+
, and a
penalty-cost function p : D R
+
. To connect client
i D to facility j, the weight w
(i, j)
of edge (i, j) is to
be paid, and to open facility j F, the opening facil-
ity cost f
j
is to be paid. In each step t, a client i D
arrives; OPC-NFL asks to either pay the penalty as-
sociated with i or connect i to an open facility. The
goal is to minimize the total penalties, the total facil-
ity opening costs, and the total connecting costs paid.
To the best of our knowledge, no online algorithm
with non-trivial competitive ratio exists for OPC-
NFL. There only is an online algorithm for the metric
version, in which facilities and clients reside in a met-
ric space and all distances respect the triangle inequal-
ity, as in (Felice et al., 2015). The latter is essential
to prove the competitive ratio of the algorithm and so
the result does not carry over to OPC-NFL.
OPC-NFL generalizes the Online Set Cover prob-
lem (OSC), due to (Alon et al., 2003), and this im-
plies an (logm logn) lower bound on the competi-
tive ratio of any online randomized polynomial-time
algorithm for OPC-NFL, where m is the number of
facilities and n is the number of clients.
In this paper, we propose the first online al-
gorithm for OPC-NFL, with asymptotically optimal
O(logm logn)-competitive ratio, where m is the num-
ber of facilities and n is the number of clients. Our al-
gorithm is randomized and is based on reducing OPC-
NFL to the Online Non-metric Facility Location prob-
lem (ONFL), due to (Alon et al., 2006).
Outline. The rest of the paper is structured as fol-
lows. In Section 3, we give an overview of results
related to our problems. In Sections 4, 5, and 6,
we present our results for OPC-CDS, OPC-VC, and
OPC-NFL, respectively. We conclude with a discus-
sion of our results and open problems in Section 7.
3 STATE-OF-THE-ART
(Qian and Williamson, 2011) initiated the study of
online prize-collecting Steiner problems by provid-
ing an O(logn)-competitive algorithm for the On-
line Prize-collecting Steiner Tree problem (OPC-ST).
(Hajiaghayi et al., 2014) proposed an online algorithm
with the same competitive ratio but gave a simpler
analysis. (Hajiaghayi et al., 2014) proposed a generic
technique that reduces online prize-collecting Steiner
problems to their corresponding fractional non-prize-
collecting variants, by losing logarithmic factor in
the competitive ratio. This has implied O(log
3
n)-
competitive and O(log
4
n)-competitive randomized
ICEIS 2021 - 23rd International Conference on Enterprise Information Systems
746
algorithms for the Online Prize-collecting Node-
weighted Steiner Tree problem (OPC-NWST) and the
Online Prize-collecting Node-weighted Steiner For-
est problem (OPC-NWSF), respectively. (Hajiaghayi
et al., 2013) gave a primal-dual algorithm with opti-
mal O(log n)-competitive ratio for ONWST in graphs
excluding a fixed graph as minor (which include pla-
nar graphs). (Hajiaghayi et al., 2014) extended this
result to OPC-NWST in graphs excluding a fixed
graph as minor, for which they gave an O(log
2
n)-
competitive algorithm. Other well-known optimiza-
tion problems were also studied in the online prize-
collecting model, such as (Ausiello et al., 2008).
(Hamann et al., 2018) proposed an online ran-
domized algorithm for a special case of OPC-CDS
in which all penalties are set to infinity, the Online
Connected Dominating Set problem (OCDS). They
showed that their algorithm has asymptotically opti-
mal O(log
2
n)-competitive ratio, where n is the num-
ber of nodes. (Markarian and Kassar, 2020) later pro-
posed an online deterministic algorithm for the prob-
lem with the same O(log
2
n)-competitive ratio.
(JunFeng and JianHua, 2014) gave an optimal 2-
approximation algorithm for the prize-collecting vari-
ant of Vertex Cover (in the offline setting). (Demange
and Paschos, 2005) studied an online model of Vertex
Cover, that is substantially different than the one in
this paper, providing competitive ratios characterized
by the maximum degree of the graph.
(Alon et al., 2006) proposed an online randomized
algorithm for a special case of OPC-NFL in which
all penalties are set to infinity, the Online Non-metric
Facility Location problem (ONFL). They showed that
their algorithm has O(log mlog n)-competitive ratio,
where m is the number of facilities and n is the
number of clients. The latter is asymptotically op-
timal since ONFL generalizes OSC. A related prob-
lem to ONFL is the metric variant, known as the On-
line Facility Location problem (OFL), in which fa-
cilities and clients reside in a metric space and all
distances respect the triangle inequality. The latter
is commonly used in competitive analysis. (Meyer-
son, 2001) gave an O(log n)-competitive randomized
algorithm for OFL, where n is the number of clients.
This was improved to an O(
logn
loglog n
)-competitive al-
gorithm by (Fotakis, 2008), who showed that this is
the best competitive ratio achievable for the problem.
Another paper of (Fotakis, 2007) gave a simple deter-
ministic primal-dual O(log n)-competitive algorithm
for the problem. The prize-collecting metric variant,
the Online Prize-collecting Facility Location prob-
lem was studied by (Felice et al., 2015), who pro-
posed an O(log n)-competitive algorithm for the prob-
lem. Their algorithm is based on previous algorithms
of (Fotakis, 2008) and (Nagarajan and Williamson,
2013).
4 ONLINE PRIZE-COLLECTING
CONNECTED DOMINATING
SET (OPC-CDS)
In this section, we present a randomized online algo-
rithm for OPC-CDS and analyze its competitive ratio.
4.1 Online Algorithm
The algorithm has two phases. In the first phase, we
transform the given instance I into an OSC instance I
0
as follows.
Given an instance I of OPC-CDS containing a
connected graph G = (V, E), a penalty cost function
p : V R
+
, and a sequence of disjoint subsets of V
arriving over time. We construct an instance I
0
of OSC
as follows. The elements of I
0
are the nodes of V .
Each node u V is represented by two sets:
a set containing u and all nodes adjacent to u, with
cost w
u
, the weight associated with u
a set containing u, with cost p
u
, the penalty asso-
ciated with u
When a subset D
t
V arrives at step t, the algorithm
returns the set P
t
containing the nodes of D
t
whose
penalties are paid for and the set CDS
t
that contains
a connected dominating set of the remaining nodes
D
t
\ P
t
. Now, the algorithm runs the algorithm for
OSC, due to (Alon et al., 2003), on I
0
and adds the
corresponding nodes to the sets P
t
and CDS
t
based on
the sets returned by the algorithm. Note that a node
may end up covered by more than one set, meaning
that its penalty might be paid for in addition to being
dominated.
Note that, any OSC solution for I
0
of cost c is a
solution of the same cost c for Phase 1 of the algo-
rithm. Moreover, an O(logm logn)-competitive algo-
rithm for OSC implies an O(log
2
n)-competitive al-
gorithm for Phase 1 of the algorithm, since the num-
ber of sets in I
0
is double the number of nodes in I
(m = 2n).
In the second phase, we connect the dominating
set nodes constructed in the first phase directly. We
run the randomized algorithm for the Online Node-
weighted Steiner Tree problem (OPC-NWST) due to
(Hajiaghayi et al., 2014) on these nodes. The two
phases of the algorithm are depicted below.
Algorithmic View of Online Prize-collecting Optimization Problems
747
Online Algorithm for OPC-CDS.
Input: G = (V, E) and subset D
t
V
Output: P
t
CDS
t
1. Run the OSC algorithm on I
0
. Add the nodes
whose penalties are paid for to P
t
and the domi-
nating set nodes to a set S
t
. If t = 1, assign any of
the nodes in S
t
as a root node r. Add all the nodes
in S
t
to CDS
t
.
2. Run the OPC-NWST algorithm to construct a tree
that connects all the nodes in S
t
to r. Add all the
nodes of this tree, that are not already in CDS
t
, to
CDS
t
.
4.2 Competitive Analysis
We denote by Opt the cost of an optimal solution Opt
I
for an instance I of OPC-CDS and by C
1
and C
2
the
cost of the two phases of the algorithm, respectively.
Phase 1. The cost C
1
of Phase 1 of the algorithm
can be bounded as follows.
C
1
O(log
2
n) · Opt
Phase 2. The cost C
2
of Phase 2 of the algorithm
is the cost of the Steiner tree nodes connecting the
dominating set nodes constructed in the first phase.
Let Opt
St
be the cost of a minimum Steiner tree of
these nodes. Since the algorithm for OPC-NWST,
due to (Hajiaghayi et al., 2014), has an O(log
2
n)-
competitive ratio, we have that C
2
O(log
2
n)·Opt
St
.
It remains to compare Opt
St
to the cost of the opti-
mal solution Opt. The latter is not a feasible solution
for Phase 2 of the algorithm. This would have been
the case in the offline setting.
Remark. In the offline setting, algorithms that first
find a dominating set and then run a Steiner tree algo-
rithm to connect the dominating set, have a straight-
forward approximation analysis, depending on the
analysis of the Steiner tree and Dominating Set al-
gorithms themselves. This means that the approxi-
mation bounds attained, in such algorithms, depend
on the approximation bounds for Dominating Set/Set
Cover and Steiner Tree (see (Guha and Khuller,
1998)).
To compare Opt
St
to Opt, we construct a Steiner
tree S for the nodes of Phase 1 of the algorithm, as fol-
lows. S will contain the nodes in the optimal solution
and some additional nodes. We add to S:
all the nodes in the optimal solution
min
iV
x
i
w
i
+
eE
p
e
y
e
Subj to: e = (i, j) E : x
i
+ x
j
+ y
e
1
i V, e E : x
i
, y
e
0 max
eE
z
e
Subj to: i V :
eγ(i)
z
e
w
i
e E : z
e
p
e
e E : z
e
0
Figure 1: LP Formulation of OPC-VC.
all the nodes added in Phase 2 of the algorithm, in
addition to the nodes added in Phase 1 (these are
the terminals and so have weight 0 each)
one additional node from the demand set D
t
, for
any t (this has weight at most w
max
)
The cost of S is upper bounded by: Opt +C
2
+ w
max
.
Thus, Opt
St
is at most Opt +C
2
+ w
max
. Therefore,
C
2
O(logn) · (Opt +C
2
+ w
max
).
Applying asymptotic notation with simple algebra
and using the fact that Opt is at least w
min
, we con-
clude that
C
2
O(logn) ·
w
min
w
max
By adding the two costs C
1
and C
2
of the algorithm,
we conclude the following theorem.
Theorem 1. There is an online O(
w
max
w
min
log
2
n)-
competitive randomized algorithm for the Online
Prize-collecting Connected Dominating Set problem
(OPC-CDS), where n is the number of nodes, w
max
is
the maximum node weight, and w
min
is the minimum
node weight.
5 ONLINE PRIZE-COLLECTING
VERTEX COVER (OPC-VC)
In this section, we present a deterministic online algo-
rithm for OPC-VC and analyze its competitive ratio.
5.1 Online Algorithm
The algorithm is a classical primal-dual algorithm.
The LP formulation of OPC-VC is depicted in Fig-
ure 1. x
i
is the indicator variable set to 1 if node i of
weight w
i
belongs to the solution, and set to 0 other-
wise. y
e
is the indicator variable set to 1 if the penalty
p
e
of edge e is paid, and set to 0 otherwise. γ(i) is the
set of edges incident to node i.
The primal-dual algorithm is depicted below. Let
S be the set of all edges whose penalties are paid for
and all nodes that are purchased by the algorithm.
ICEIS 2021 - 23rd International Conference on Enterprise Information Systems
748
Online Algorithm for OPC-VC.
Input: G = (V, E) and e E
Output: S
1. Increase the dual variable z
e
of e until one of the
dual constraints becomes tight.
2. Set the primal variable corresponding to each tight
constraint to 1.
3. Purchase each node corresponding to a tight con-
straint and pay each penalty corresponding to a
tight constraint.
5.2 Competitive Analysis
Let S be the primal solution constructed by the al-
gorithm. Recall that S is the set of all edges whose
penalties are paid for and all nodes that are purchased
by the algorithm. We have that the dual constraint cor-
responding to each node i S is tight: w
i
=
eγ(i)
z
e
.
Moreover, the dual constraint corresponding to each
e S is tight: p
e
= z
e
. Thus,
eS
p
e
y
e
eE
z
e
and
iS
x
i
w
i
=
iS
eγ(i)
z
e
2 ·
eE
z
e
.
By the Weak Duality theorem, we have that
eE
z
e
Opt, where Opt is the cost of the optimal
solution, and hence the theorem follows.
Theorem 2. There is an online 3-competitive de-
terministic algorithm for the Online Prize-collecting
Vertex Cover problem (OPC-VC).
6 ONLINE PRIZE-COLLECTING
NON-METRIC FACILITY
LOCATION (OPC-NFL)
In this section, we present a randomized online algo-
rithm for OPC-NFL and analyze its competitive ratio.
6.1 Online Algorithm
Given an instance I of OPC-NFL that contains a
complete bipartite graph G = ((F D), E), an edge-
weight function w : E R
+
, a facility-opening-cost
function f : F R
+
, and a penalty-cost function
p : D R
+
. The algorithm is based on transforming
I into an instance I
0
of the Online Non-metric Facility
Locatiom problem (ONFL), as follows.
We add to the set F, a facility j and set its opening
cost to 0.
For each client i D that arrives, we add an edge
from i to j and set its weight to the penalty cost of
i.
The algorithm is depicted below.
Online Algorithm for OPC-NFL.
Input: G = ((F D), E) and instance I of OPC-NFL
Output: Set of penalties, facility costs, and connect-
ing costs paid
1. Transform I into I
0
, as described earlier.
2. Run the algorithm for ONFL on I
0
.
3. Purchase all facilities and edges outputted by the
ONFL algorithm. For each arriving client i, pay
its associated penalty if the corresponding edge in
I
0
is purchased by the ONFL algorithm.
6.2 Competitive Analysis
Let I be the original instance of OPC-NFL. Let Opt
be an optimal solution of I and let C
Opt
be its cost.
Let I
0
be the new instance of ONFL generated from I
as above. Let Opt’ be an optimal solution of I
0
and let
C
Opt
0
be its cost.
We need to show that Opt is a feasible solution
of I
0
: Given a client i, whenever its penalty is pur-
chased in Opt, we purchase the corresponding edge
in I
0
; whenever a facility is opened in Opt, we open
it too in I
0
and whenever an edge is paid for, we pay
for it too in I
0
. This means that every time a client
arrives, it is connected to at least one facility and the
connecting edge is paid for by the solution Opt. Thus,
Opt is a feasible solution of I
0
. Hence, C
Opt
0
C
Opt
,
since every feasible solution is lower bounded by the
cost of the optimal solution.
The algorithm for ONFL has an O(logm
0
logn
0
)-
competitive ratio, where m
0
is the number of facilities
and n
0
is the number of clients. According to our re-
duction, m
0
= m+1 and n
0
= n, where m is the number
of facilities and n is the number of clients in the orig-
inal instance I.
Let C be the cost of our solution for I. Our solution
is constructed by running the algorithm for ONFL and
thus C O(log m log n) ·C
Opt
0
O(log m log n) ·C
Opt
and the theorem below follows.
Theorem 3. There is an online asymptotically opti-
mal O(log m log n)-competitive randomized algorithm
for the Online Prize-collecting Non-metric Facility
Location problem, where m is the number of facilities
and n is the number of clients.
Algorithmic View of Online Prize-collecting Optimization Problems
749
7 DISCUSSION AND OPEN
PROBLEMS
In this paper, we have studied OPC-CDS in general
graphs. Connected Dominating Set problems have
been extensively studied in the offline setting. These
were motivated by real-world network applications in
which geometric graph models were used (see (Mah-
dian and Yan, 2011; Amb
¨
uhl et al., 2006)). One re-
search direction is to extend this study to the online
setting by considering such graphs for OPC-CDS and
its variants. This might yield to competitive ratios
dependent on the properties of the geometric graphs
rather than the number of nodes.
Another set of open problems would generate by
considering other online models. We have made our
study based on the oblivious adversary model. It is
interesting to consider other weaker adversary models
such as stochastic as in (Manshadi et al., 2010), or
random as in (Mahdian and Yan, 2011).
Competitive analysis offers a worst case perfor-
mance evaluation of online algorithms. It would be
interesting to also consider other models. (Cheung,
2016) performed a computational study of various on-
line algorithms for Steiner problems to reveal their
performance in average. It would be interesting to do
the same for our proposed algorithms, as in (Hamann
et al., 2018)), for the Online Connected Dominating
Set problem. Another study by (Angelopoulos, 2019)
based the analysis of online algorithms on additional
parameters of the problem, known as parameterized
analysis of online algorithms. For the Online Node-
weighted Steiner Tree problem, he showed a tight
competitive ratio that depends on the maximum node
weight, minimum node weight, and the number of ter-
minals. Investigating parameterized analysis for our
problems would initiate an interesting study.
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